Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Symmetrizations of Ball-Bodies

Shiri Artstein-Avidan, Dan I. Florentin

TL;DR

The paper investigates symmetrization procedures within the ball-body class ${\cal S}_n$, emphasizing the interaction between the $c$-duality and symmetrizations. It develops linear parameter systems based on the $c$-hull ${\rm conv}_c$, proving concavity of ${\rm Vol}(L_t^c)^{1/n}$ along these paths and establishing planar convexity results for ${\rm Vol}(L_t)$, with higher-dimensional convexity failing in general. It shows Steiner symmetrization increases the dual volume and preserves the ball-body class in the plane, but constructs a concrete $3$-D counterexample where the Steiner symmetral of a ball-body leaves ${\cal S}_3$, and demonstrates that arbitrarily flat counterexamples exist for $n\ge3$. Together, these results delineate a sharp dimensional boundary for preservation under Steiner symmetrization and highlight the limits of symmetry-improvement strategies in the ball-body setting.

Abstract

We study symmetrization procedures within the class $\mathcal S_n$ of \emph{ball-bodies}, i.e.\ intersections of unit Euclidean balls (equivalently, summands of the Euclidean unit ball, or $c$-convex sets via the $c$-duality $A\mapsto A^c$). We first examine linear parameter systems obtained by replacing the usual convex hull by the $c$-hull $A^{cc}$, deriving consequences for volume along these $c$-paths. In particular, we obtain convexity statements in special cases and in dimension $2$, and we show by example that such convexity fails in general for $n\ge 3$. We then focus on Steiner symmetrization. We prove that Steiner symmetrization increases the \emph{dual volume} and that in the planar case Steiner symmetrals of ball-bodies remain ball-bodies. In contrast, we provide an explicit example in $\RR^3$ showing that the Steiner symmetral of a ball-body need not belong to $\mathcal S_n$, and show that there are such counter-examples with arbitrarily large curvatures.

Symmetrizations of Ball-Bodies

TL;DR

The paper investigates symmetrization procedures within the ball-body class , emphasizing the interaction between the -duality and symmetrizations. It develops linear parameter systems based on the -hull , proving concavity of along these paths and establishing planar convexity results for , with higher-dimensional convexity failing in general. It shows Steiner symmetrization increases the dual volume and preserves the ball-body class in the plane, but constructs a concrete -D counterexample where the Steiner symmetral of a ball-body leaves , and demonstrates that arbitrarily flat counterexamples exist for . Together, these results delineate a sharp dimensional boundary for preservation under Steiner symmetrization and highlight the limits of symmetry-improvement strategies in the ball-body setting.

Abstract

We study symmetrization procedures within the class of \emph{ball-bodies}, i.e.\ intersections of unit Euclidean balls (equivalently, summands of the Euclidean unit ball, or -convex sets via the -duality ). We first examine linear parameter systems obtained by replacing the usual convex hull by the -hull , deriving consequences for volume along these -paths. In particular, we obtain convexity statements in special cases and in dimension , and we show by example that such convexity fails in general for . We then focus on Steiner symmetrization. We prove that Steiner symmetrization increases the \emph{dual volume} and that in the planar case Steiner symmetrals of ball-bodies remain ball-bodies. In contrast, we provide an explicit example in showing that the Steiner symmetral of a ball-body need not belong to , and show that there are such counter-examples with arbitrarily large curvatures.
Paper Structure (8 sections, 21 theorems, 50 equations, 2 figures)

This paper contains 8 sections, 21 theorems, 50 equations, 2 figures.

Table of Contents

  1. Introduction and preliminaries
  2. Symmetrizations of Ball-Bodies
  3. Linear parameter systems
  4. The planar case
  5. Steiner symmetrizations and the class ${\cal S}_n$
  6. Steiner Symmetrization of ball-bodies in the plane
  7. A counterexample in dimension $3$
  8. How flat can counterexamples get in dimension $n\ge 3$

Key Result

Lemma 2.1

Let $u\in S^{n-1}, K\in {\cal S}_n$. Then $M_u K\in{\cal S}_n$, and $M_u(K^c) = (M_u K)^c$. Thus the notation $M_u K^c$ is unambiguous, and moreover ${\rm Vol}(M_u K^c) \ge {\rm Vol}(K^c)$.

Figures (2)

  • Figure 1: Illustrating the functions
  • Figure 2: Two planar lenses

Theorems & Definitions (39)

  • Lemma 2.1
  • Corollary 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • proof
  • Theorem 2.5: Bezdek
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 29 more